1. **State the problem:** We need to simplify the function $$f(x) = \frac{4x^4 - 16}{6x^2 + 7x - 3}$$. 2. **Recall the formulas and rules:** - Factor polynomials by taking out common factors or using special products. - Simplify rational expressions by factoring numerator and denominator and canceling common factors. 3. **Factor the numerator:** $$4x^4 - 16 = 4(x^4 - 4) = 4(x^2 - 2)(x^2 + 2)$$ 4. **Factor the denominator:** We factor $$6x^2 + 7x - 3$$ by finding two numbers that multiply to $$6 \times (-3) = -18$$ and add to 7. These numbers are 9 and -2. Rewrite: $$6x^2 + 9x - 2x - 3 = 3x(2x + 3) -1(2x + 3) = (3x - 1)(2x + 3)$$ 5. **Rewrite the function with factored forms:** $$f(x) = \frac{4(x^2 - 2)(x^2 + 2)}{(3x - 1)(2x + 3)}$$ 6. **Check for common factors:** There are no common factors between numerator and denominator to cancel. 7. **Final simplified form:** $$f(x) = \frac{4(x^2 - 2)(x^2 + 2)}{(3x - 1)(2x + 3)}$$ This is the simplified form of the function.